How to connect 2 vertices with edges?
The answer is simple. Just press F between two vertices.

And your tetrahedron problem can be solved as follows:


Start with a Plane.
Delete one vertex and join all vertices with F to make a triangle.
Subdivide the triangle.
Go to the outer vertices and scale them on the xy axis to 0.
In the last step you only have to move the selected vertices on the Z axis by one.




That's it.

(Of course, don't forget to swap normals.)